Hausdorff Dimension and Brownian Motion

This lecture covers the concept of Hausdorff dimension applied to the set of instants where Brownian motion is zero, showing that the dimension is 1/2. The instructor explains the construction of coverings for the set of zeros, demonstrating that the dimension is less than 1/2. Through examples involving intervals and Cantor sets, the lecture illustrates how to estimate the dimension of sets with non-integer values. The application of the Hausdorff dimension to Brownian motion leads to the theorem that the set of zeros has a dimension of 1/2 with probability 1. The lecture concludes with a proof that the dimension is less than 1/2 with probability 1 for a specific interval, setting the stage for further exploration.